Spectral localization, power boundedness and invariant subspaces under Ritt's type condition

• Autorzy: Yu. Lyubich
• Języki publikacji: EN

Abstrakty

For a bounded linear operator T in a Banach space the Ritt resolvent condition $∥R_λ(T)∥ ≤ C/|λ - 1|$ (|λ| > 1) can be extended (changing the constant C) to any sector |arg(λ - 1)| ≤ π - δ, $arccos(C^{-1}) < δ < π/2$. This implies the power boundedness of the operator T. A key result is that the spectrum σ(T) is contained in a special convex closed domain. A generalized Ritt condition leads to a similar localization result and then to a theorem on invariant subspaces.

Kategorie tematyczne

• 47A35: Ergodic theory
• 47A10: Spectrum, resolvent

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1999
• Tom: 134
• Numer: 2
• Strony: 153-167

Daty

wydano

1999

otrzymano

1998-03-16

poprawiono

1998-12-17

Twórcy

autor

Yu. Lyubich

• Department of Mathematics, Technion, 32000 Haifa, Israel

Bibliografia

• [1] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, 1962.
• [2] Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), 313-328.
• [3] Yu. Lyubich and V. Matsaev, Operators with separable spectrum, in: Amer. Math. Soc. Transl. (2) 47 (1965), 89-129.
• [4] B. Nagy and J. Zemánek, A resolvent condition implying power boundedness, Studia Math. 134 (1999), 143-151.
• [5] O. Nevanlinna, Convergence of Iterations for Linear Equations, Birkhäuser, 1993.
• [6] R. K. Ritt, A condition that $lim_{n→∞}n^{-1}T^n = 0$, Proc. Amer. Math. Soc. 4 (1953), 898-899.
• [7] J. G. Stampfli, A local spectral theory for operators IV; Invariant subspaces, Indiana Univ. Math. J. 22 (1972), 159-167.
• [8] E. Tadmor, The resolvent condition and uniform power boundedness, Linear Algebra Appl. 80 (1986), 250-252.